A241405 -id:A241405 - cp's OEIS frontend

A323757 Modified exponential perfect numbers: numbers k such that A241405(k) = 2*k.

6, 60, 90, 264, 3960, 8736, 87360, 131040, 1868160

Offset: 1

Amiram Eldar, Jan 26 2019

Each term of this sequence corresponds to a primitive e-perfect number (A054980, see formula and Andrew Lelechenko's comment in A241405).

Also in the sequence are 1028004440830371164160, 20546724596095746048000, and 146361946186458562560000 (corresponding to the 3 additional terms of A054980 given by Andrew Lelechenko). - Amiram Eldar, Jul 18 2019

A. V. Lelechenko, The Quest for the Generalized Perfect Numbers, Theoretical and Applied Aspects of Cybernetics, Proceedings, The 4th International Scientific Conference of Students and Young Scientists, Kyiv, 2014.
David Moews, Perfect, amicable and sociable numbers.

Cf. A003557, A054980, A241405.

f[p_, e_] := DivisorSum[e+1, p^(#-1)&]; mesigma[1]=1; mesigma[n_] := Times @@ f @@@FactorInteger@n; mePerfectQ[n_] := mesigma[n]==2n; Select[Range[10000], mePerfectQ]

f(n) = {my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1)));} \\ A241405
isok(n) = f(n) == 2*n; \\ Michel Marcus, Jan 30 2019

a(n) = A003557(A054980(n)).

A379029 Modified exponential abundant numbers: numbers k such that A241405(k) > 2*k.

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 114, 120, 138, 150, 168, 174, 186, 210, 222, 246, 258, 270, 282, 294, 318, 330, 354, 366, 390, 402, 420, 426, 438, 462, 474, 498, 510, 534, 546, 570, 582, 606, 618, 630, 642, 654, 660, 678, 690, 714, 726, 750, 762, 770, 780, 786, 798, 822

Offset: 1

Amiram Eldar, Dec 14 2024

All the squarefree abundant numbers (A087248) are terms since A241405(k) = A000203(k) for a squarefree number k.
If k is a term and m is coprime to k them k*m is also a term.
The numbers of terms that do no exceed 10^k, for k = 2, 3, ..., are 5, 67, 767, 7595, 76581, 764321, 7644328, 76468851, 764630276, ... . Apparently, the asymptotic density of this sequence exists and equals 0.07646... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000203, A241405, A323757, A323758, A323759, A379027.
Subsequence of A005101.
Subsequences: A034683, A087248, A379030, A379031.
Similar sequences: A064597, A129575, A129656, A292982, A348274, A348604.

f[p_, e_] := DivisorSum[e + 1, p^(# - 1) &]; mesigma[1] = 1; mesigma[n_] := Times @@ f @@@ FactorInteger[n]; meAbQ[n_] := mesigma[n] > 2*n; Select[Range[1000], meAbQ]

is(n) = {my(f=factor(n)); prod(i=1, #f~, sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1))) > 2*n;}

A379031 Odd modified exponential abundant numbers: odd numbers k such that A241405(k) > 2*k.

Original entry on oeis.org

15015, 19635, 21945, 23205, 25935, 26565, 31395, 33495, 33915, 35805, 39585, 41055, 42315, 42735, 45885, 47355, 49665, 50505, 51765, 54285, 55965, 58695, 61215, 64155, 68145, 70455, 72345, 77385, 80535, 82005, 83265, 84315, 91245, 95865, 102795, 112035, 116655

Offset: 1

Amiram Eldar, Dec 14 2024

First differs from its subsequences A112643 and A249263 at n = 51: a(51) = 195195 is not a term of these two sequences.
First differs from its subsequence A129485 at n = 363: a(363) = 2537535 is not a term of A129485.
First differs from A339938 at n = 28: A339938(28) = 75075 is not a term of this sequence.
First differs from A360526 at n = 46: A360526(46) = 165165 is not a term of this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A005408 and A379029.
Subsequence of A005231.
Subsequences: A112643, A129485, A249263.
Cf. A241405.
Similar sequences: A094889, A127666, A129485, A293186, A321147, A339938, A348275, A360526.

f[p_, e_] := DivisorSum[e + 1, p^(# - 1) &]; mesigma[1] = 1; mesigma[n_] := Times @@ f @@@ FactorInteger[n]; meAbQ[n_] := mesigma[n] > 2*n; Select[Range[1, 10^5, 2], meAbQ]

is(k) = if(!(k%2), 0, my(f=factor(k)); prod(i=1, #f~, sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1))) > 2*k);

A379030 Nonsquarefree modified exponential abundant numbers: nonsquarefree numbers k such that A241405(k) > 2*k.

Original entry on oeis.org

120, 150, 168, 270, 294, 420, 630, 660, 726, 750, 780, 840, 924, 990, 1014, 1020, 1050, 1092, 1140, 1170, 1320, 1380, 1386, 1428, 1470, 1530, 1560, 1596, 1638, 1650, 1710, 1734, 1740, 1848, 1860, 1890, 1950, 2040, 2058, 2070, 2142, 2166, 2184, 2220, 2280, 2394

Offset: 1

Amiram Eldar, Dec 14 2024

All the squarefree abundant numbers (A087248) are also modified exponential abundant numbers (A379029). This sequence lists the terms of A379029 that are not in A087248.

The numbers of terms that do no exceed 10^k, for k = 3, 4, ..., are 14, 211, 2090, 21236, 212744, 2123071, 21235175, 212450318, ... . Apparently, the asymptotic density of this sequence exists and equals 0.0212... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Intersection of A013929 and A379029.
Complement of A087248 within A379029.
Cf. A005117, A241405.

f[p_, e_] := DivisorSum[e + 1, p^(# - 1) &]; mesigma[1] = 1; mesigma[n_] := Times @@ f @@@ FactorInteger[n]; nsmeAbQ[n_] := !SquareFreeQ[n] && mesigma[n] > 2*n; Select[Range[2400], nsmeAbQ]

is(n) = {my(f=factor(n)); if(issquarefree(f), 0, prod(i=1, #f~, sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1))) > 2*n);}

A379032 Numbers k such that k and k+1 have an equal sum of modified exponential divisors: A241405(k) = A241405(k+1).

Original entry on oeis.org

14, 44, 957, 1334, 1485, 1634, 1652, 2204, 2685, 3195, 3451, 3956, 4136, 5547, 8495, 8636, 8907, 9844, 11515, 12256, 14876, 15608, 19491, 20145, 20155, 27519, 27643, 33998, 35235, 36575, 38180, 41265, 41547, 42818, 45716, 48364, 74918, 79316, 79826, 79833, 84134

Offset: 1

Amiram Eldar, Dec 14 2024

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1918 (terms below 10^10)

Cf. A241405.

Similar sequences: A002961, A064115, A064125, A293183, A306985.

f[p_, e_] := DivisorSum[e + 1, p^(# - 1) &]; mesigma[1] = 1; mesigma[n_] := mesigma[n] = Times @@ f @@@ FactorInteger[n]; Select[Range[10^5], mesigma[#] == mesigma[#+1] &]

mesigma(n) = {my(f=factor(n)); prod(i=1, #f~, sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1))); }
lista(kmax) = {my(m1 = 1, m2); for(k = 2, kmax, m2 = mesigma(k); if(m1 == m2, print1(k-1, ", ")); m1 = m2);}

A379027 Irregular table read by rows in which the n-th row lists the modified exponential divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 2, 3, 6, 1, 7, 1, 2, 8, 1, 9, 1, 2, 5, 10, 1, 11, 1, 3, 4, 12, 1, 13, 1, 2, 7, 14, 1, 3, 5, 15, 1, 16, 1, 17, 1, 2, 9, 18, 1, 19, 1, 4, 5, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 3, 6, 8, 24, 1, 25, 1, 2, 13, 26, 1, 3, 27, 1, 4, 7, 28

Offset: 1

Amiram Eldar, Dec 14 2024

If the prime factorization of n is Product_{i} p_i^e_i, then the modified exponential divisors of n are all the divisors of n that are of the form Product_{i} p_i^b_i such that 1 + b_i | 1 + e_i for all i.

			The table starts:
  1;
  1, 2;
  1, 3;
  1, 4;
  1, 5;
  1, 2, 3, 6;
  1, 7;
  1, 2, 8;
  1, 9;
  1, 2, 5, 10;
  1, 11;
  1, 3, 4, 12;

Amiram Eldar, Table of n, a(n) for n = 1..11627 (first 2000 rows)
Andrew V. Lelechenko, The Quest for the Generalized Perfect Numbers, Theoretical and Applied Aspects of Cybernetics, Proceedings, The 4th International Scientific Conference of Students and Young Scientists, Kyiv, 2014.
David Moews, Perfect, amicable and sociable numbers.

Cf. A379028 (row lengths), A241405 (row sums).

Similar tables: A027750 (all divisors), A077609 (infinitary), A077610 (unitary), A222266 (bi-unitary), A322791 (exponential), A361255 (exponential unitary).

modexpDivQ[n_, d_] := Module[{f = FactorInteger[n]}, And @@ MapThread[Divisible, {f[[;; , 2]] + 1, IntegerExponent[d, f[[;; , 1]]] + 1}]]; row[1] = {1}; row[n_] := Select[Divisors[n], modexpDivQ[n, #] &]; Table[row[n], {n, 1, 28}] // Flatten

ismodexpdiv(f, d) = {my(e); for(i=1, #f~, e = valuation(d, f[i, 1]); if((f[i, 2]+1) % (e+1), return(0))); 1; }
row(n) = {my(f = factor(n), d = divisors(f), mediv = [1]); if(n == 1, return(mediv)); for(i=2, #d, if(ismodexpdiv(f, d[i]), mediv = concat(mediv, d[i]))); mediv; }

A241757 Numbers n such that 2n is a sum of two primes, the adding of which requires only one carry in binary.

Original entry on oeis.org

2, 11, 15, 23, 27, 29, 39, 45, 47, 51, 55, 57, 59, 63, 71, 77, 87, 95, 99, 103, 105, 107, 111, 115, 117, 119, 123, 125, 127, 131, 135, 137, 143, 147, 149, 155, 159, 165, 171, 173, 175, 177, 179, 183, 185, 187, 189, 191, 197, 203, 207, 215, 219, 221, 223, 225

Offset: 1

Vladimir Shevelev, Apr 28 2014

nonn

Apart from a(1), both primes are 1 mod 4, hence 2 is the only even term in the sequence. - Charles R Greathouse IV, Apr 29 2014

			2 is in the sequence since 2*2=2+2 is a sum of two primes and adding 2+2 requires only one carry in binary.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A241405.

is(n)=if(n%2==0, return(n==2)); forprime(p=2,n,if(p%4==1 && isprime(2*n-p) && bitand(p, 2*n-p)==1, return(1))); 0 \\ Charles R Greathouse IV, Apr 29 2014

MSB(n)=2^(#binary(n)-1);
is(n)={
    if(n%2==0, return(n==2));
    my(V=(n - MSB(n))>>1, k=0);
    while(k=bitand(k-V,V),    \\ Note: assignment, not comparison
        my(p=4*k+1,q=2*n-p);
        if(isprime(p) && isprime(q), return(1))
    );
    0
}; \\ Charles R Greathouse IV, Apr 30 2014

More terms from Peter J. C. Moses, Apr 29 2014

A241758 Smallest prime in representation 2*A241757(n) by sum of two primes, the adding of which in binary requires only one carry.

Original entry on oeis.org

2, 5, 13, 5, 17, 17, 5, 17, 5, 5, 13, 17, 5, 13, 5, 17, 37, 17, 5, 13, 17, 17, 29, 37, 41, 5, 5, 17, 13, 5, 13, 17, 5, 37, 41, 17, 5, 73, 5, 89, 13, 97, 5, 13, 17, 37, 41, 29, 137, 5, 5, 41, 5, 41, 13, 193, 5, 5, 17, 193, 17, 17, 37, 41, 37, 97, 53, 73, 53, 5

Offset: 1

Vladimir Shevelev, Apr 28 2014

			a(2)=5, since A241757(2)=22=5+17, and in binary in sum of 101+10001 involves only one carry.

Peter J. C. Moses, Table of n, a(n) for n = 1..10000
Aviezri Fraenkel and Alex Kontorovich, The Sierpiński Sieve of Nim-varieties and Binomial Coefficients, INTEGERS 7 (2)(2007), #A14.
E. E. Kummer, Über die Ergänzungssätze zu den allgemeinen Reciprocitätsgesetzen, J. Reine Angew. Math. 44 (1852), 93-146.

Cf. A241757, A241405.

2||Binomial(2*A241757(n), a(n)). Indeed, from the Kummer theorem (see reference) 2^t||Binomial(n,x) if and only if in adding x and n-x in binary we have exactly t carries. A proof of the Kummer theorem in arbitrary base one can find in [Fraenkel & Kontorovich].

More terms from Peter J. C. Moses, Apr 29 2014

A323758 Lesser of modified exponential amicable pairs.

Original entry on oeis.org

114, 366, 1140, 3660, 3864, 5016, 11040, 16104, 16536, 44772, 57960, 67158, 68640, 142290, 142310, 180960, 196248, 198990, 240312, 248040, 308220, 322080, 326424, 339822, 348840, 352632, 366792, 462330, 669900, 671580, 785148, 815100, 817440, 849240, 912072

Offset: 1

Amiram Eldar, Jan 26 2019

nonn

A modified exponential amicable pair (m, n) has A241405(m) = A241405(n) = m + n + 1.

The larger counterparts are in A323759.

Amiram Eldar, Table of n, a(n) for n = 1..4725 (terms below 2*10^11)
David Moews, Perfect, amicable and sociable numbers.

Cf. A241405, A323757, A323759.

f[p_, e_] := DivisorSum[e+1, p^(#-1)&]; mesigma[1]=1; mesigma[n_] := Times @@ f @@@FactorInteger@ n; s={}; mes[n_] := mesigma[n]-n; Do[m=mes[n]; If[m>n && mes[m]==n, AppendTo[s, n]], {n,1,10000}]; s

f(n) = {my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1)));} \\ A241405
fm(n) = f(n) - n;
isok(n) = my(fn=fm(n)); (fn > n) && (fm(fn) == n); \\ Michel Marcus, Jan 30 2019

A323759 Larger of modified exponential amicable pairs.

Original entry on oeis.org

126, 378, 1260, 3780, 4584, 5544, 11424, 16632, 16728, 49308, 68760, 73962, 88608, 179118, 168730, 212160, 225096, 256338, 266568, 250920, 365700, 374304, 391656, 374418, 387720, 386568, 393528, 548550, 827700, 739620, 827652, 932100, 912288, 935400, 1052088

Offset: 1

Amiram Eldar, Jan 26 2019

nonn

A modified exponential amicable pair (m, n) has A241405(m) = A241405(n) = m + n + 1.

The lesser counterparts are in A323758.

Amiram Eldar, Table of n, a(n) for n = 1..4725 (terms whose lesser counterparts below 2*10^11)
David Moews, Perfect, amicable and sociable numbers.

Cf. A241405, A323757, A323758.

f[p_, e_] := DivisorSum[e+1, p^(#-1)&]; mesigma[1]=1; mesigma[n_] := Times @@ f @@@FactorInteger@ n; s={}; mes[n_] := mesigma[n]-n; Do[m=mes[n]; If[m>n && mes[m]==n, AppendTo[s, m]], {n,1,10000}]; s

f(n) = {my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2]+1, d, f[i, 1]^(d-1)));} \\ A241405
fm(n) = f(n) - n;
isok(n) = my(fn=fm(n)); (fn > 0) && (fn < n) && (fm(fn) == n); \\ Michel Marcus, Jan 30 2019

cp's OEIS Frontend

A323757 Modified exponential perfect numbers: numbers k such that A241405(k) = 2*k.

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A379029 Modified exponential abundant numbers: numbers k such that A241405(k) > 2*k.

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A379031 Odd modified exponential abundant numbers: odd numbers k such that A241405(k) > 2*k.

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A379030 Nonsquarefree modified exponential abundant numbers: nonsquarefree numbers k such that A241405(k) > 2*k.

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A379032 Numbers k such that k and k+1 have an equal sum of modified exponential divisors: A241405(k) = A241405(k+1).

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A379027 Irregular table read by rows in which the n-th row lists the modified exponential divisors of n.

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A241757 Numbers n such that 2n is a sum of two primes, the adding of which requires only one carry in binary.

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A241758 Smallest prime in representation 2*A241757(n) by sum of two primes, the adding of which in binary requires only one carry.

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